Abstract In this paper we give a lower bound for the discriminant of the polynomials { G k , i ( x ) } defined by G k , 0 ( x ) = 1 , G k , 1 ( x ) = x + 1 and G k , i + 2 ( x ) = x G k , i + 1 ( x ) − ( k − 1 ) G k , i ( x ) f o r i ≥ 0 . These polynomials are closely related to the Chebyshev polynomials of the second kind. We derive our result by using orthogonality properties of the polynomials { F k , i ( x ) } defined by F k , 0 ( x ) = 1 , F k , 1 ( x ) = x , F k , 2 ( x ) = x 2 − k and F k , i + 2 ( x ) = x F k , i + 1 ( x ) − ( k − 1 ) F k , i ( x ) f o r i ≥ 1 .

中文翻译：

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与 Chebyshev 多项式相关的多项式的判别式的下限

摘要 在本文中，我们给出了由 G k , 0 ( x ) = 1 , G k , 1 ( x ) = x + 1 和 G k 定义的多项式 { G k , i ( x ) } 的判别式的下界, i + 2 ( x ) = x G k , i + 1 ( x ) − ( k − 1 ) G k , i ( x ) fori ≥ 0 。这些多项式与第二类切比雪夫多项式密切相关。我们通过使用由 F k , 0 ( x ) = 1 , F k , 1 ( x ) = x , F k , 2 ( x ) = 定义的多项式 { F k , i ( x ) } 的正交性来推导出我们的结果x 2 − k 和 F k , i + 2 ( x ) = x F k , i + 1 ( x ) − ( k − 1 ) F k , i ( x ) fori ≥ 1 。